Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

GRYTCZUK, A., On, some connections between Legendre symbols and continued fractions

On some connections between Legendre symbols and continued fractions ALEKSANDER GRYTCZUK Abstract. In this note we give a complement of some results of Friesen given in [2] about some connections between Legendre symbols and continued fractions. 1. Introduction In the paper [1] P. Chowla and S. Chowla gave several conjectures concerning continued fractions and Legendre symbols. Let d — pq, where p,q are primes such that p = 3 (mod 4), q = 5 (mod 8) and let y/d = [go

; q\

, .. ., q

sj be the representation of y/d as a simple continued fraction. s Denote by S = Then P. Chowla and S. Chowla conjectured the i=i following relationship: = (— l)

s, where is the Legendre's symbol. This conjecture has been proved by A. Schinzel in [3]. Further interesting results for d = pq = 1 (mod 4) and for d — 2pq was given by C. Friesen in [2]. From his results summarized in the Table 1 on page 365 of [2] it follows that in the following cases: p ~ 3 (mod 8),q = 1 (mod 8) or p = 7 (mod 8), q = 1 (mod 8) or p = 1 (mod 8), q = 3 (mod 8) or p = 1 (mod 8), q = 7 (mod 8) are not known a connection between Legendre's symbol and the representation of ^/pq as a simple continued fraction. In this connection we prove the following Theorem: Theorem. Let d — pq ~ 3 (mod 4) and yfpq — [qo; q\

, .. ., q

s], then s = 2m; c

m = 2, p, q; and Í?) = if

C

m = p .where c

m is defined by the following recurrent formulas: , b

m + b

m

+i = c

mq

m, d = pq = b

2

m+

1 + c

mc

m+

i. q

m qo + b

7